Drawing a card- four cards are drawn from a desk without replacement.Find these probabilities.

Mathematics

1 answer:

svetlana [45]3 years ago

5 0

The deck consists of 4 jacks, 26 black cards, and 13 hearts.

There is only 1 way of drawing all 4 jacks: $\dbinom44=\dfrac{4!}{4!(4-4)!}=1$ .

There are $\dbinom{26}4=14,950$ ways of drawing 4 black cards.

There are $\dbinom{13}4=715$ ways of drawing 4 hearts.

There are $\dbinom{52}4=270,725$ ways of drawing any 4 cards.

The probabilities are then

$\dfrac{\binom44}{\binom{52}4}=\dfrac1{207,725}\approx0.00000369$

$\dfrac{\binom{26}4}{\binom{52}4}=\dfrac{46}{833}\approx0.0552$

$\dfrac{\binom{13}4}{\binom{52}4}=\dfrac{11}{4165}\approx0.00264$

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a) P(X < 99) = 0.2033.

b) P(98 < X < 100) = 0.4525

Step-by-step explanation:

To solve this question, we need to understand the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$ , the z-score of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .